I would echo what ironclad said, though. It's true we don't know the whole of it and as Dan said above, there are degrees to these things, but I would say that trust CAN be rebuilt. Sure, it won't be easy, but it is possible. If he is truly remorseful, there are also certain "safeguards" that you can agree to to keep something like this from happening again.

You'll just have to walk through the next few weeks together and both decide (together and separately) where you want to be. It won't be easy and you'll be in my thoughts/prayers during this tough time.

I will let you know that I have seen marriages put back together and thrive after things like this, but if he isn't willing to change and isn't willing to work to build that trust back up, then that probably isn't possible.